Run VSE movie strip output through Compositor or Compositor movie clip output through VSE within one project possible?
You can make a separate Compositor Scene that only outputs the compositor result. Then in the VSE scene you can add the Compositor scene as a scene strip.

You can not send the output of the VSE anywhere except for mixdown to codec. Don't worry, everyone asks this at some point.

The VSE is designed to be the end of the production chain.